The spectral description of turbulence allows us to decompose velocity and pressure fields in terms of wavenumbers and frequencies, or length and time scales. We discuss the notion of scale decomposition and introduce several properties of the Fourier transform between physical (spatial/temporal) space and scale (spectral) space in various dimensions, including complex conjugate relations for real functions and Parseval’s theorem. The Fourier transform allows us to develop useful relations between correlations and energy spectra, which are used extensively in the statistical theory of turbulence. The one-dimensional and three-dimensional energy spectra are specifically discussed in conjunction with Taylor’s hypothesis to enable spectra computation from single-point time-resolved measurements. The discrete version of the transform, or the discrete Fourier series, is then introduced, as it is typically encountered in numerical simulations and postprocessing of discrete experimental data. Treatment of periodic data is first considered, followed by nonperiodic data with the help of windowing. The procedure for the computation of various discrete spectra is outlined.

This chapter, along with the next three, covers the general topic of clock generation and distribution as well as clock and data synchronization. Clocking circuitry, including clock and data recovery systems can dissipate 30–50% of total transceiver power. Jitter degrades BER as much as amplitude noise. Therefore wireline designers must pay attention to the jitter performance of clock generation circuitry and clock distribution circuitry as much as they focus on the amplitude noise behaviour of the receiver’s signal path. Although clock generation can dissipate non-trivial power, the centralized generation of a clean reference clock allows the amortization of its power dissipation across multiple transceiver lanes, although its distribution is also challenging. Therefore, a thorough treatment of all aspects of clocking is important. This chapter gives an overview of the principal synchronization approaches used in wireline systems as well as clock distribution circuitry.

An overview of the three modern categories of methods for numerical prediction of turbulent flows is provided: direct numerical simulation (DNS), solution of the Reynolds-averaged Navier-Stokes (RANS) equations, and large-eddy simulation (LES). We describe zero-equation, one-equation, two-equation, and Reynolds stress transport models for the RANS equations. RANS computations require significantly fewer grid points and lower computational cost since the solutions are smooth and turbulent structures are not captured, but there is a need to tune model parameters for different flows to match experimental data. In LES, only the large-scale motions are resolved, whereas unresolved small scales are modeled. We introduce the notion of filtering, subgrid-scale parameterization, as well as the seminal dynamic Smagorinsky subgrid-scale model. Wall-resolved and wall-modeled LES are briefly discussed. With ever increasing computer power, as well as advances in numerical methods and subgrid-scale models, LES is rapidly becoming a viable tool for practical computations. In selecting a method, one should consider quantities to be predicted, accuracy of the predictions, and the computational cost.

In this chapter we introduce circuits that are used to generate clocks. After an introduction to the metrics of oscillators, this chapter introduces the main categories of oscillators, and how they are made voltage-controlled. The important trade-offs between power, phase-noise, and tuning range are explained. Methods for converting a VCO to a digitally controlled oscillator are included in Section 14.4.

Electrical-link design is challenging due to the frequency-dependent loss and reflections associated with electrical channels as well as cross-talk between nearby channels. The equations describing lossless and lossy transmission lines are introduced in this chapter, followed by a brief discussion of loss mechanisms. The characteristics of various channels are presented, along with the effect of wirebonds and packages. The goal of this chapter is to provide link designers with a methodology to estimate the overall pulse response of a channel consisting of a lossy transmission line and the relevant package and chip parasitic elements. Knowing the pulse response, a link designer can then contemplate and model the equalization (discussed in Chapter 4) needed to properly detect transmitted bits. This chapter is organized to discuss transmission-line fundamentals and then overall channels.

Turbulent flow is an important branch of fluid mechanics with wide-ranging occurrences and applications, from the formation of tropical cyclones to the stirring of a cup of coffee. Turbulence results in increased skin friction and heat transfer across surfaces, as well as enhanced mixing. As such, it is of practical significance, and there is a need to establish predictive methods to quantify turbulent flows. Equally important is a physical understanding of turbulent flows to guide strategies to model and control turbulence-driven phenomena. We focus on the study of turbulent flows and draw on theoretical developments, experimental measurements, and results from numerical simulations. Turbulent flows are governed by the Navier-Stokes equations. The solution of these equations for turbulent flows displays chaotic and multiscale behavior. When averaged, the nonlinear terms in the Navier-Stokes equations lead to the so-called closure problem, where additional unknowns are introduced in the mean flow equations. These unknowns are typically modeled using intuition, experience, and dimensional arguments. We present the scaling and dimensional analysis necessary for model development.

Mixed-signal wireline receivers use sensitive, high-speed decision circuits to compare input signals to a well-defined threshold, allowing the decoding of received bits. This chapter gives the important metrics of these circuits and contrasts their operation against that of conventional D flip-flops. Two main classes of circuit are discussed, namely current-mode logic and sense-amplifier-based circuits. The basics of their operation are presented. Simulation strategies and offset compensation are discussed.

This chapter starts with an overview of the requirements of an electrical-link transmitter. Electrostatic discharge protection is required in CMOS processes. The large capacitance it adds severely loads the output driver. The main mitigation strategies are presented here, focusing on T-coil-based compensation, a type of inductive peaking. In Section 1.12, two categories of circuits were introduced, namely CML and CMOS. These give rise to current-mode and voltage-mode transmitters, both of which are presented in this chapter. Impedance control and the implementation of feed-forward equalizers is discussed. A reader who is primarily interested in optical transceiver design should read this chapter before moving on to Chapter 8.

A central challenge in the design of electrical links is to compensate for frequency-dependent loss in the channel that introduces inter-symbol interference (ISI). This chapter presents the overall objectives of joint Tx/Rx equalization. The system-level operation of transmitter-side feed-forward equalizers (FFEs) is discussed. Circuit details are presented in Chapter 5. Receiver-side continuous-time linear equalizers (CTLEs) and finite-impulse-response (FIR) filters are discussed next, followed by decision-feedback equalizers (DFEs). DFEs differ from FFEs, CTLEs and FIRs in that they only remove ISI rather than attempt to invert the low-pass channel characteristic. With the growing trend toward ADC- based receivers, the implementation of DFEs and Rx FFEs is discussed in the analog domain and the digital domain. The topics in this chapter are also a relevant background for the sections in Chapter 10 that discuss TIAs for reduced bandwidth systems, where equalization is used to remove ISI from an intentionally bandwidth-limited optical receiver front-end.

This chapter builds on the basic transmitter building blocks presented in Chapter 5 and applies them to optical links. The section on direct modulation discusses CML and SST circuits for driving laser diodes, along with how equalization can be added, incorporating concepts from Chapter 4. One aspect of laser diodes that is presented is their nonlinear behaviour that favours having different equalization for rising and falling edges, something not typically done in electrical links where channels are linear. Circuit design for modulator drivers is discussed in Section 8.2, including an overview of distributed amplifiers. Finally, microring modulator drivers are presented in Section 8.3. Since the basic IC building blocks have already been introduced in Chapter 5, this chapter emphasizes aspects particular to driving optical devices, such as the electrical modelling of the electronics/photonic packaging and the challenges in producing voltage swings beyond the breakdown voltage of the underlying CMOS technology.

Virtually all technologically relevant applications involve interactions of turbulent flows with solid walls, including flows over aircrafts and automobiles. We study these interactions using canonical wall-bounded flows, including fully developed channels, pipes, and flat-plate boundary layers, with a focus on channel flow. A common scaling may be employed in the near-wall region using the friction velocity and viscous length scale to derive the so-called wall units. In this region, which comprises the viscous sublayer, buffer layer, and overlap layer, the law of the wall governs the mean velocity profile, and the constant-stress-layer assumption is often employed. We discuss key features of the mean velocity profile, particularly the log law in the overlap region, which stands as a celebrated result in turbulence theory. Away from the wall, the outer layer scales with the boundary-layer thickness and freestream velocity. We discuss the skin friction and wake laws to describe the mean outer velocity profile and introduce the Clauser chart method. We also examine in more detail the scales and structural features of turbulence near a wall, including streaks and hairpin vortices.

This chapter introduces wireline communication, focusing on introducing key terminology, such as eye diagrams, intersymbol interference, noise, bit-error ratio, pulse response, non-return-to-zero modulation, pulse-amplitude modulation. The similarities and differences between electrical and optical links are discussed. Exemplary transceiver block diagrams are presented.

In turbulent free-shear flows, fluid streams interact to generate regions of turbulence that evolve without being limited or confined by solid boundaries. Such interactions create mean shear, which is a source of turbulent kinetic energy that results in enhanced flow mixing. Far downstream, the flow retains little memory of its origins and exhibits self-similar behavior. Its mean velocity profile, turbulence intensities, and Reynolds stresses, when scaled appropriately, become independent of downstream distance as it freely expands into its surroundings. Free-shear flows occur in combustors, vehicle wakes, and jet engine exhaust. We focus our attention on three canonical categories of such flows: jets, wakes, and mixing layers. A detailed similarity analysis of the plane jet is provided alongside summarized results for the plane wake and mixing layer. We introduce examples involving turbines in wind farms and drag on wake-generating bodies. The notion of entrainment, which is central to the expansion of free-shear flows, is discussed. We also examine the scales and structural features of turbulent free-shear flows, including streamwise rib vortices and spanwise rollers.

This chapter begins with a recapitulation of an optical link and what the general requirements of an optical receiver are. The discussion of optical receivers starts with a brief analysis of a passive current-voltage converter (i.e., a resistor) in terms of its gain, bandwidth and input-referred noise. This section proposes reasonable bandwidth requirements for optical receivers that do not use equalizers, so-called low-ISI systems. Open-loop and feedback amplifiers are considered. Additional amplification through main-amplifier design is explained, starting with the effect on bandwidth of cascading multiple first-order stages. Behaviour of second- and third-order systems are also presented. Examples of Cherry-Hooper, second-order active feedback and third-order active feedback as well as interleaving feedback are presented. CMOS inverter-based designs are discussed.